As U.S. trade representative Robert Lighthizer rushes to amend the North American Free Trade Agreement as soon as the end of next week, key questions have emerged: Could President Donald Trump enact changes to Nafta without Congress? Would he try to do so?

The answers are complex. Under the U.S. Constitution, the president has the right to negotiate treaties. But Congress sets tariffs, and Congress often agrees to outsource much of those powers to the executive branch, particularly to the Office of the U.S. Trade Representative. Those powers are typically granted on condition the administration consults with and seeks the consent of Congress, and adheres to procedural rules.

President Donald Trump hasn’t hidden his impatience with the rules. “We have these provisions where you have to wait long periods of time, you have to notify Congress, and after you notify Congress, you have to get certified,” Mr. Trump said in a speech a year ago. “The whole thing is ridiculous.”

Mr. Trump has also claimed the legal authority to pull out of Nafta, a threat he has made repeatedly as part of his negotiating tactics. Many members of Congress say the president doesn’t have such authority.

The president’s team has followed procedural rules while negotiating, according to congressional staffers. While the president can make minor changes to the treaty on his own authority, any change that affects U.S. law would require the approval of Congress. It is unclear whether the current update of Nafta, under negotiation with Canadian and Mexican officials behind closed doors, will result in a deal that would require a change in federal law, say people following the talks.

Asked about the congressional strategy for Nafta on Monday, White House spokeswoman Sarah Sanders said, “Those conversations and negotiations are ongoing.”

The biggest priority for the Trump administration—tightening Nafta’s rules for auto trade in ways it hopes will boost American employment in the industry—doesn’t require changes to U.S. law and could be enacted without Congress, experts say. The administration's proposal to scale back or eliminate investor-state dispute settlement, an arbitration system popular with international businesses, also wouldn’t need congressional approval. But its push to end another dispute-settlement system, in Nafta’s Chapter 19, would change U.S. law, according to a business group following the talks.

While the Trump administration hasn’t spelled out its exact plan for implementing changes it is seeking to Nafta, the comments so far suggest that as a procedural matter, a deal could soon go to Congress under a 2015 law known as fast track, or trade promotion authority, which allows the president to receive an up-or-down vote on his trade plans. Congress gave the president that right in 2015. It expires in July 2021 if Congress doesn’t block an automatic renewal by this July.

Some congressional and business advisers say they’re analyzing the possibility that Messrs. Trump and Lighthizer could scale back the ambition of the negotiations to achieve a quick agreement—a deal that might not need congressional approval—before Mexico’s presidential election and the U.S. midterm elections.

The U.S. law that implemented Nafta more than two decades ago gives a president room to unilaterally proclaim changes to tariffs and other Nafta rules under certain circumstances. But Mr. Trump would have to consult with key congressional committees on these changes, according to the congressional Research Service.

Changing Nafta without congressional cooperation poses risks for the administration. Republican committee leaders have rejected some of Mr. Trump’s trade priorities. And it could alienate Democratic lawmakers who are currently supportive of Mr. Trump’s approach to trade, potentially generating opposition to the president’s party during the 2018 midterm elections.